Almost half a century ago Charles Elton (1958) warned of the increasing frequency of foreign species introduction, and of the inevitable biological dislocations that follow.  Today, the number and type of invading organisms is growing --- understanding and monitoring the process of alien species spread is an important problem in mathematical ecology.  A key element of this process is prediction of spread rate for the invader.  It was thought for many years that this issue of spread rate was essentially resolved by analysis of a nonlinear partial differential  equation derived by Fisher (1937) for invading  genotypes.  It is now clear that the Fisher spread model does not hold for many relevant biological situations. 

First, the model tacitly ignores rare, long distance dispersal events that initiate secondary invasion foci, far ahead of the bulk of invasion.  These events can be shown to drive the invasion process at much higher speeds than previously thought, at speeds which may continue to increase as the invasion progresses.  The resulting spatial pattern of spread is patchy, with the patches linked historically via long-distance dispersal. I will discuss some new mathematical results for integrodifference equations (discrete-time maps in continuous space) that can be used to predict such patterns.

Second, the model does not include secondary ecological interactions with resident species.  When these interactions are competitive, interesting problems arise in the calculation of the spread rate.   I will discuss some of these in the context of the general theory for systems of integrodifference equations.